Cohomology, deformations and extensions of averaging Hom-associative algebras

S. Braiek, S. Mabrouk

In this paper, we present the concept of averaging Hom-associative algebras, which are Hom-associative algebras endowed with an averaging operator. We define representations and dual representations for these algebras without imposing any specific conditions. Subsequently, we develop a cohomology theory tailored to averaging Hom-associative algebras. We further explore both infinitesimal and formal deformation theories, demonstrating that the introduced cohomology serves as the deformation cohomology. Additionally, we define abelian extensions of averaging Hom-associative algebras and establish that the equivalence classes of such extensions correspond to elements in the second cohomology group. 

Advanced Studies: Euro-Tbilisi Mathematical Journal, Vol. 19(2) (2026), pp. 195-212